On New Connectivity Topological Indices of Certain Chemical Graphs

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Vol. 71 No. 3s3 (2022) 250 http://philstat.org.ph

On New Connectivity Topological Indices of Certain Chemical Graphs

Mohamad Nazri Husin¹, Muhammad Sharjeel², Ghulam Farid², Zainab Javid², Khurrem Shehzad², Murat Cancan³, Mehdi Alaeiyan⁴, Mohammad Reza Farahani⁴

1Special Interest Group on Modelling & Data Analytics (SIGMDA), Faculty of Ocean Engineering Technology & Informatics,

Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia.

[email protected]

2 Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan.

[email protected], [email protected], [email protected],

3 Faculty of Education, Van Yuzuncu Yıl University, Zeve Campus, Tuşba, 65080, Van, Turkey.

[email protected], [email protected].

4 Department of Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran.

[email protected], [email protected], [email protected] Correspondence should be addressed to Mohamad Nazri Husin; [email protected]

(ORCID ID: 0000-0003-4196-4984, 0000-0002-0019-9131, 0000-0002-8606-2274, 0000-0003-2185-5967, 0000-0003-2969-4280)

Correspondence should be addressed to: [email protected]

Issue: Special Issue on Mathematical Computation in Combinatorics and Graph Theory in Mathematical Statistician and Engineering Applications

Article Info

Page Number: 250 - 261 Publication Issue:

Vol 71 No. 3s3 (2022)

Article History

Article Received: 30 April 2022 Revised: 22 May 2022

Accepted: 25 June 2022 Publication: 02 August 2022

Abstract

The goal of our research is to calculate degree-based connectivity Kulli- Basava indices of chemical graphs. These graphs include hammer-like- benzenoid and phenylene graphs. Several degree-based topological indices are calculated of hammer and phenylene graph.

Keywords: Degree-based Topological index, benzenoid graphs, sum connectivity index, product connectivity index, atom-bond connectivity index, geometric arithmetic connectivity index, reciprocal connectivity index.

2010 Mathematics Subject Classification: 05C07, 05C09, 05C31, 5C76, 05C99.

1. Introduction

Cheminformatics is a modern field that combines chemistry, computer science, mathematics.

Mathematical chemistry is a subfield of chemistry in which we examine chemical structures by applying mathematical logic. The molecular graph is constructed by representing atoms as points or vertices and constraints as edges or lines, respectively. A graph is ‘joined’ if every two vertices have connectivity, a graph is ‘simple’ if there are no multiple edges and loops present between any two vertices. Molecular graphs are all connected and simple. The number of vertices connected to a vertex determines its degree. We recommend [4] for basic graph theory concepts. A topological index is a numerical value associated with a graph that describes the graph's topology and is invariant under graph auto-morphism. Distance-based

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topological indices, degree-based topological indices, counting-related polynomials, graph indices are some of the most common types of topological indices. Degree-based topological indices are significantly vital in chemical graph theory, especially in chemistry. A topological index 𝑇𝑜𝑝(𝐺^′) of a graph 𝐺^′ is a value having the characteristic that 𝑇𝑜𝑝(𝐻^′) = 𝑇𝑜𝑝(𝐺^′) for any graph 𝐻^′isomorphic to 𝐺^′. Wiener developed the concept of the topological index while researching the boiling point of paraffin [1, 3, 9-11, 24, 31]. For more information about the topological indices, the reader can look at the articles [2, 7, 8, 17-23, 26-28, 30, 36-38, 40]

2. Definitions

For a graph G’, the Sum connectivity Kulli-Basava index, the Product connectivity Kulli- Basava index, the Atom bond connectivity Kulli-Basava index, the Geometric arithmetic connectivity Kulli-Basava index, the Reciprocal connectivity Kulli-Basava index are defined by

𝑆𝐾𝐵(𝐺^′) = ∑ 1

√𝑆𝑒( 𝑢) + 𝑆_𝑒( 𝑣)

𝑢𝑣∈𝐸(𝐺)

, (1)

𝑃𝐾𝐵(𝐺^′) = ∑ 1

√𝑆𝑒( 𝑢) 𝑆_𝑒( 𝑣)

, (2)

𝐴𝐵𝐶𝐾𝐵(𝐺^′) = ∑ √𝑆_𝑒( 𝑢) + 𝑆_𝑒( 𝑣) − 2 𝑆𝑒( 𝑢) 𝑆𝑒( 𝑣)

, (3)

𝐺𝐴𝐾𝐵(𝐺^′) = ∑ 2√𝑆_𝑒( 𝑢) 𝑆_𝑒( 𝑣) 𝑆_𝑒( 𝑢) + 𝑆_𝑒( 𝑣)

, (4) and 𝑅𝐾𝐵(𝐺^′) = ∑_{𝑢𝑣∈𝐸(𝐺)}√𝑆𝑒( 𝑢) 𝑆_𝑒( 𝑣), (5) respectively (see [33]).

Figure 1: Hammer-like benzenoid graph 𝐻_𝑛(𝐺^′).

The degree of an edge is determined by the formula𝑑_𝐺(𝑒) = d_𝐺(𝑢) + 𝑑_𝐺(𝑣) − 2. The neighbourhood of vertices and sum of the degrees of all edges incident to a vertex are denoted by 𝑁_𝐺(𝑣) and𝑆_𝑒(𝑣), respectively (see [5-6, 12-16, 25, 29, 32-35, 39]).

3. Main Results

3.1 Result for Hammer like-Benzenoid Graph

The hammer-like benzenoid graph H Gn

( )

 has 2(2n+15) vertices and (5n+37) edges. By simplification, the basis of the degree of vertices is divided into two portions 2(n+8) vertex of degree 2 and 2(n+7) vertex of degree 3. When two pyrene-fragments are joined at each

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end of a linear polyacene chain of length n, the resulting structure is called a hammer-like structure Hn

( )

G ^.

Now, we construct a table by using section Error! Reference source not found. and calculate the partition of the edges to their sum of the degrees of all edges incident to vertices for the hammer-like benzenoid graph. So, for^e⁼^uv^^{E G}

( )

, we have:

Table 1: The number of edges partition of (𝑺_𝒆(𝒖), 𝑺_𝒆(𝒗)) for Hammer like-Benzenoid Graph.

(S u_e( ), S ( ))_e v (4,5) (5,5) (5,10) (5,11) (6,10) (6,11) Number of

edges ^{n n}

(

⁻²

)

² ^{n n}

(

⁻²

)

ⁿ ^{n n}

(

⁻¹

)

ⁿ

(S u_e( ), S ( ))_e v (10,12) (11,11) (11,12) (12,12) (10,10) Number of

edges

n n+2 n 2 n−1

Theorem 3.1.1. The Sum connectivity Kullibasava index H Gn

( )

is defined by

( )

¹ ¹ ¹ ² ¹ ¹ ² ¹ ² ²

3 15 4 17 20 22 23 3 15

2 2 1 1

10 22 6 20 .

SKB G =  + + n + + + + − − n

   

 

+ + + − 

 

Proof: By using sections Error! Reference source not found. andTable 1, we get

( )

^{( )}

( ) ( )

= 1

uv E G e e

SKB G

S u S v

 





+

( ) ( ) ( ) ( )

( )

1 1 1 1 1 1 1

2 2 2 1 1

9 10 15 16 16 17 20

1 1 1 1

2 2

22 22 23 24

n n n n n n n n n

n n n

= − + + − + + − + + −

+ + + + +

( ) ( ) ( ) ( )

( )

1 1 1 1 1 1 1

2 2 2 1 1

3 10 15 4 4 17 20

1 1 1 1

2 2

22 22 23 2 6

n n n n n n n n n

n n n

= − + + − + + − + + −

+ + + + +

1 1 1 2 1 1 2 1 2 2

3 15 4 17 20 22 23 3 15

2 1 2 1

.

10 6 22 20

n n

   

= + +  + + + + − − 

   

 

+ + + − 

Theorem 3.1.2. The Product connectivity Kullibasava index Hn

( )

G is given as

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( )

¹ ¹ ¹ ² ¹ ¹ ¹ ¹ ¹ ¹ ² ² ¹

10 11

20 50 60 55 66 120 132 20 50 60

2 2 1 1 5 11 6 10 .

PKB G = + + n + + + + + + − − − n

 

+ + + − 

Proof: By using section Error! Reference source not found. and Table 1, we get

( )

^{( )}

( ) ( )

1

uv E G e e

PKB G

S u S v

 

 =



( ) ( ) ( )

( ) ( )

2 2 2

2

1 1 1 1 1 1

2 2 2

20 25 50 55 60 66

1 1 1 1 1

1 2 2

100 120 121 132 144

1 1 1 2 2 1 1

5 11 6 10

20 50 60

1 1 1 1 1 1 2 2 1

10 11 .

55 66 120 132 20 50 60

n n n n n n n n

n n n n

n

= − + + − + + − +

+ − + + + + +

   

=  + +  + + + − 

 

+ + + + + + − − − 

Theorem 3.1.3. The Atom bond connectivity Kullibasava index for Hn

( )

G is given as

( )

⁷ ¹³ ⁷ ² ^{2 8} ²² ^{2 20} ¹⁸

20 50 30 5 6 11 10

ABCK G  n  

 = + +  + + + − 

   

14 15 18 1 20 21 7 7 13

2 2 .

55 66 10 6 11 132 20 30 50 n

 

+ + + + + + − − − 

Proof: It follows from section Error! Reference source not found. and Table 1that

( ) ( ) ( )

( ) ( )

( )

e e 2

uv E G e e

S u S v ABCKB G

S u S v





+ −

 =



( ) ( ) ( )

( ) ( )

7 8 13 14 14 15

2 2 2 1

20 25 50 55 60 66

18 20 20 21 22

1 2 2

100 120 121 132 144

n n n n n n n n

n n n n

= − + + − + + − +

+ − + + + + +

( ) ( ) ( )

( ) ( )

2 7 8 2 13 14 2 7 15

2 2 2

20 5 50 55 30 66

18 1 20 21 22

1 2 2

10 6 11 132 12

n n n n n n n n

n n n n

= − + + − + + − +

+ − + + + + +

7 13 7 2 2 8 22 2 20 18

20 50 30 5 6 11 10

14 15 18 1 20 21 7 7 13

2 2 .

55 66 10 6 11 132 20 30 50

n

   

= + +  + + + − 

   

 

+ + + + + + − − − 

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Theorem 3.1.4. The Geometric arithmetic connectivity Kullibasava index for Hn

( )

G is defined by

( )

2

2 55 66 1 120 2 132 20 2 50 60

55 33 5 60 11 66 5 25 30

20 50 60 4 4 1 1

10 25 30 5 11 3 5 .

GAKB G n

n

 

 = + + + + + − − − 

   

+ + +  + + + −  Proof: By using section (2) and Table 1, we get

( ) ( ) ( )

( ) ( )

( )

2 _e _e

uv E G e e

S u S v GAKB G

S u S v

 

 =



+

( ) ( ) ( )

( ) ( )

2 20 4 25 2 50 2 55 2 60

2 2 1

20 25 50 55 60

2 66 2 100 2 120 2 121 2 132 4 144

1 2

66 100 120 121 132 144

n n n n n n n

n n n n n

= − + + − + + −

+ + − + + + + +

( ) ( ) ( )

( ) ( )

20 20 50 2 55 60 66

2 2 1

10 25 25 55 30 33

1 120 2 132 1

1 2

5 60 11 66 3

n n n n n n n n

n n n n

= − + + − + + − +

+ − + + + + +

2

2 55 66 1 120 2 132 20 2 50 60

55 33 5 60 11 66 5 25 30

20 50 60 4 4 1 1

10 25 30 5 11 3 5 .

n n

 

= + + + + + − − − 

 

   

+ + +  + + + − 

Theorem 3.1.5. We define the Reciprocal connectivity Kullibasava index for Hn

( )

G by

( ) ( ) ( )

( )

7 15 2 17 20 2 22 23 6 2 15 2 10 2 22 2 24 20 .

RKB G = + n + + + + − − n

+ + + −

Proof: By using section Error! Reference source not found.andTable 1, we get

( ) ( ) ( )

( ) ^e ^e

uv E G

RKB G S u S v





 =



( ) ( ) ( )

( ) ( )

2 4 5 2 5 5 2 5 10 5 11 1 6 10

6 11 1 10 10 10 12 2 11 11 11 12 2 12 12

n n n n n n n

n n n n n

= − + + + + − + + + + − +

+ + + − + + + + + + + + + +

( ) ( ) ( ) ⁽ ⁾

( )

2 2 2

2 3 2 10 2 15 4 4 17 1 20

22 23 2 24 2 22

n n n n n n n n n

n n n

= − + + − + + − + + −

+ + + + +

( ) ( )

( )

3 4 15 2 6 2 15 4 4 17 20 22 23 22

2 10 2 22 2 24 20

n n

= + + + − − + − + + + + +

+ + + −

(

⁷ ¹⁵

) (

ⁿ² ¹⁷ ²⁰ ^{2 22} ^{23 6 2 15}

) (

ⁿ ^{2 10} ^{2 22} ^{2 24} ^{20 .}

)

= + + + + + − − + + + −

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3.2 Results for Linear Benzenoid Phenylenes

The descriptor phenylene is generally reserved for chemical graphs formed by inserting a square between all pairs of neighbouring hexagons in catacondensed benzenoid systems.

There are two types of vertices one vertex is of degree Two and the other vertex is of degree Three. The number of vertices is equal to 𝑉(𝐺^′) = ℎ^′2+ 𝑞^′2+ 𝑛^′2+ ℎ^′𝑞^′+ 𝑞^′𝑛^′− 5and the total number of edges in the graph 𝐸(𝐺^′) = ℎ^′2+ 𝑞^′2+ 𝑛^′2+ 5(ℎ^′+ 𝑞^′+ 𝑛^′) − 4, where ℎ^′is the number of hexagons in each hexagon segment 𝑞^′is the number of squares joining the hexagon and 𝑛^′ is the total number of hexagons segments. Therefore, linear benzenoid phenylenes can be a suitable term for these graphs.

Figure 2: Linear Benzenoid Phenylene where ℎ^′= 3, 𝑞^′= 5 and 𝑛^′ = 4.

Now, we construct the table using section 2 and we calculate the partitions of the edges to their sum of the degrees of all edge’s incident to vertices of fluoranthene type benzenoid linear generalized phenylene graph for 𝑒 = 𝑢𝑣 ∈ 𝐸(𝐺^′) in Table 2.

Table 2: The number of edges partition of (𝑺_𝒆(𝒖), 𝑺_𝒆(𝒗)) for Linear Benzenoid Phenylene.

(S u_e( ), S ( ))_e v (4,4) (4,5) (5,10) (6,6) (6,10) Number of

edges

2 n n q +1 q²

(S u_e( ), S ( ))_e v (6,11) (10,10) (11,11) (11,12) (12,12) Number of

edges

h+ +q n q q +1 h+ +q n h²+n²+5

Theorem 3.2.1. The Sum connectivity Kulli basava index for linear generalized phenylene is defined by

( )

²

2 2

1 1 1 1 1 1 1 1

24 17 23 12 17 20 22 23

1 1 1 1 1 1 1 1 1 5

4 24 3 15 17 23 2 12 22 24 .

SKB G h h q

q n n

   

 

 =   + +  + + + + +  

     

 

   

    

+   +  + + + +  + + + +  Proof: By using definition (2) and Error! Reference source not found., we get

( )

^{( )}

( ) ( )

= 1

uv E G e e

SKB G

S u S v

 





+

2

2 2

1 1 1 1 1 1 1

2 ( 1) ( )

8 9 15 12 16 17 20

1 1 1

( 1) ( ) ( 5)

22 23 24

n n q q h q n q

q h q n h n

       

= + + + + + + + + + +

+ + + +  +  +  +

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2

2 2

2 2 2

1 1 1 1 1 1 1

3 4

2 15 12 12 17 17 17 20

1 1 1 1 1 1 1

5

22 22 23 23 23 24 24 24

1 1 1 1 1 1 1 1

24 4 24 12 17 20 22 23

1 1 1 1 1 1

17 23 3 15 17 23

n n n

q q h q q

q h q n h n

h q n q

h

       

= + + + + + + + + +

     

+ + + + + + + +

 

         

=  +   +  + + + + + 

    

+ +  + + + + 

   

1 1 1 5

.

2 12 22 24

n + + + + 

Theorem 3.2.2. The Product connectivity Kulli basava index for linear generalized phenylene is given by

( )

¹ ² ¹ ² ¹ ² ¹ ¹ ¹ ¹ ¹

12 60 12 6 66 10 11 132

1 1 1 1 1 1 1 1 1 5

2 6 11 12

66 132 20 50 66 132

PKB G h q n q

h n

   

 =   +   +   + + + + +  

       

+ +  + + + +  + + + +  Proof: By using definition (2) and Error! Reference source not found., it follows that

( )

^{( )}

( ) ( )

1

uv E G e e

PKB G

S u S v





 =



2

2 2

1 1 1 1 1 1 1 1

2

16 20 50 36 36 60 66 66

1 1 1 1 1 1 1

66 100 121 121 132 132 132

1 1 1

5

144 144 144

n n q q h q

n q q h q n

h n

     

= + + + + + + +

     

+ + + + + + +

 

+ + +

2

1 1 1 1 1 1 1 1 1

2 120 50 6 6 60 66 66 66

1 1 1 1 1 1 1 1 1

10 11 11 132 132 132 12 12 512

n n q q h q n

q q h q n h n

      

= + + + + + + + +

      

+ + + + + + + + +

2 2 2

1 1 1 1 1 1 1 1

12 60 12 6 66 10 11 132

1 1 1 1 1 1 1 1 1 5

2 6 11 12 .

66 132 120 50 66 132

h q n q

h n

   

       

=  +  +  + + + + + 

       

+ +  + + + +  + + + + 

Theorem 3.2.3. We define the Atom bond connectivity Kullibasava index for linear generalized phenylene

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( )

² ²

2

22 7 10 15 18 20 21

12 30 6 66 10 11 132

22 15 21 7 13 15 21

12 66 132 20 50 66 132

6 10 20 5 22

2 6 11 12 .

ABCK G h q q

n h n

     

 =   +   + + + + +  

     

  

+  + +  + + + + 

 

+ + + + 

 

Proof: By using definition (2) and Error! Reference source not found., we get

( ) ( ) ( )

( ) ( )

( )

e e 2

uv E G e e

S u S v ABCKB G

S u S v

 

+ −

 =



2

2 2

6 7 13 10 14 15

2 ( 1) ( )

16 20 50 36 60 66

18 20 21 22

( 1) ( ) ( 5)

100 121 132 144

n n q q h q n

q q h q n h n

      

= + + + + + + + +

      

+ + + + + + + + +

2

2 2

6 7 13 10 10 7 15 15

2 20 50 6 6 30 66 66

15 18 20 20 21 21 21

66 10 11 11 132 132 132

22 22 5 22

12 12 12

22 7 10 15 18 20 21

12 30 6 66 10 11 132

22 12

n n q q h q

n q q h q n

h n

h q q

     

= + + + + + + +

     

+ + + + + + +

 

+ + +

     

  

=  +  + + + + + 

 

+  

2 15 21 7 13 15 21

66 132 20 50 66 132

6 10 20 5 22

2 6 11 12 .

n  h  n

 + +  + + + +  

    

 

+ + + + 

Theorem 3.2.4. The Geometric arithmetic connectivity Kulli basava index for linear generalized phenylene is given by

2 60 2 2 2 66 2 132 2 66 2 132

( ) 3

8 17 23 17 23

2 20 2 50 2 66 2 132

+ 9

9 15 17 23

GAKB G h q n h q

n

     

 =  +   +  + +  + + +  

 

+ + + +

 

 

Proof: By using definition (2) andError! Reference source not found., we get

( ) ( ) ( )

( ) ( )

( )

2 _e _e

uv E G e e

S u S v GAKB G

S u S v





 =



+

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2

2 2

16 2 20 2 50 2 36 2 60 2 66

4 ( 1) ( )

8 9 15 12 16 17

2 100 2 121 2 132 2 144

( 1) ( ) ( 5)

20 22 23 24

n n q q h q n

q q h q n h n

      

= + + + + + + + +

      

+ + + + + + + + +

2

2 2

2 20 2 50 60 2 66 2 66 2 66

2 1

9 15 8 17 17 17

2 132 2 132 2 132

1 5

23 23 23

n n q q h q n

q q h q n h n

      

= + + + + + + + +

      

+ + + + + + + + +

2 60 2 2 2 66 2 132 2 66 2 132

8 17 23 3 17 23

2 20 2 50 2 66 2 132

9 15 17 23 9.

h q n h q

n

     

    

= +  + + +  + + + 

 

+ + + +  +

 

Theorem 3.2.5. The Reciprocal connectivity Kulli basava index for linear generalized phenylene is given by

( ) ( ) ( ) ^{( )} ( ) ( )

( )

2 2 2

12 60 12 66 132 27 66 132

20 50 66 132 85.

RKB G h q n h q

n

 =  +  +  + + + + + 

+ + + + +

Proof: By using section Error! Reference source not found.andError! Reference source not found., we get

( ) ( ) ( )

( ) ^e ^e

uv E G

RKB G S u S v

 

 =



2

2 2

2 16 20 50 ( 1) 36 60 ( ) 66 100

( 1) 121 ( ) 132 ( 5) 144

n n q q h q n q

q h q n h n

       

= + + + + + + + + +

     

+ + + + + + + +

2

2 2

2.4 20 50 ( 1)(6) 60 ( ) 66 (10)

( 1)(11) ( ) 132 ( 5)(12)

n n q q h q n q

q h q n h n

       

= + + + + + + + + +

     

+ + + + + + + +

( ) ( ) ^{( )} ( ) ( )

( )

2 2 2

12 60 12 66 132 27 66 132

20 50 66 132 85.

h q n h q

n

    

= + + + + + + +

+ + + + +

Conclusion

Graph theory has given researchers several helpful tools, such as topological indices. It investigates quantitative structure-activity (QSAR) and structure-property (QSPR) correlations, which are used to determine chemical compound biological activities and characteristics. In this paper, the Kulli-Basava indices of chemical graphs have been computed. Our results may play a vital role in estimating the melting and boiling points.

Acknowledgement

The authors would like to thanks Universiti Malaysia Terengganu for the support of this research work via research vot number: P29000. The authors also thank to anonymous referees for their valuable suggestion for the improvement of the manuscript

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Vol. 71 No. 3s3 (2022) 259 http://philstat.org.ph References

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